Lies, Damned Lies, and Quantum Statistics?

Statistics has a bad reputation, and has had for a long time, as demonstrated by Mark Twain’s famous quote[1] that I paraphrased to use as the title of this blog post. Of course physics is supposed to be above the fudging of statistical numbers to make a point. Well, on second thought, theoretical physics should be above fudging (in the experimental branch, things are not so clear cut).

Statistical physics is strictly about employing all mathematically sound methods to deal with uncertainty. This program turned out to be incredibly powerful, and gave a solid foundation to the thermodynamic laws. The latter were empirically derived previously, but only really started to make sense once statistical mechanics came into its own, and temperature was understood to be due to the Brownian motion. Incidentally, this was also the field that first attracted a young Einstein’s attention. Among all his other accomplishments, his paper on the matter that finally settled the debate if atoms were for real or just a useful model is often overlooked. (It is mindboggling that within a short span 0f just 40 years (’05-’45) science went from completely accepting the reality of atoms, to splitting them and unleashing nuclear destruction).

Having early on cut his teeth on statistical mechanics, it shouldn’t come as a surprise that Einstein’s last great contribution to physics went back to this field. And it all started with fudging the numbers, in a far remote place, one that Einstein had probably never even heard of.

In the city that is now the capital of Bangladesh, a brilliant but entirely unknown scholar named Satyendra Nath Bose made a mistake when trying to demonstrate to his students that the contemporary theory of radiation was inadequate and contradicted experimental evidence. It was a trivial mistake, simply a matter of not counting correctly. What added insult to injury, it led to a result that was in accordance with the the correct electromagnetic radiation spectrum. A lesser person may have just erased the blackboard and dismissed the class, but Bose realized that there was some deeper truth lurking beneath the seemingly trivial oversight.

What Bose stumbled upon was a new way of counting quantum particles. Conventionally, if you have two particles that can only take on two states, you can model them as you would the probabilities for a coin toss. Lets say you toss two coins at the same time; the following table shows the possible outcomes:

Coin 1

Head

Tail

Coin 2

Head

HH

HT

Tail

TH

TT

It is immediate obvious that if you throw two coins the combination head-head will have a likelihood of one in four. But if you have the kind of “quantum coins” that Bose stumbled upon then nature behaves rather different. Nature does not distinguish between the states tails-head and head-tails i.e. the two states marked green in the table. Rather it just treats these two states as one and the same.

In the quantum domain nature plays the ultimate shell game. If these shells were bosons the universe would not allow you to notice if they switch places.

This means, rather than four possible outcomes in the quantum world, we only have three, and the probability for them is evenly spread, i.e. assigning a one-third chance to our heads-heads quantum coin toss.

Bose found out the hard way that if you try to publish something that completely goes against the conventional wisdom, and you have to go through a peer review process, your chances of having your paper accepted are almost nil (some things never change).

That’s where Einstein came into the picture. Bose penned a very respectful letter to Einstein, who at the time was already the most famous scientist of all time, and well on his way to becoming a pop icon (think Lady Gaga of Science). Yet, against all odds, Einstein read his paper and immediately recognized its merits. The rest is history.

In his subsequent paper on Quantum Theory of Ideal Monoatomic Gases, Einstein clearly delineated these new statistics, and highlighted the contrast to the classical one that produces unphysical results in the form of an ultraviolet catastrophe. He then applied it to the ideal gas model, uncovering a new quantum state of matter that would only become apparent at extremely low temperatures.

His audacious work set the state for the discovery of yet another fundamental quantum statistic that governs fermions, and set experimental physics on the track to achieving ever lower temperature records in order to find the elusive Bose-Einstein condensate.

This in turn gave additional motivation to the development of better particle traps and laser cooling. Key technologies that are still at the heart of the NIST quantum simulator.

UPDATE: For some reason Because this site got slashdottednew comments are currently not showing up in my heavily customized WordPress installation – I get to see them in the admin view and can approve them but they are still missing here.

My apologies to everybody who took the time to write a comment! Like most bloggers I love comments so I’ll try to get this fixed ASAP.

“lies, damned lies and statistics” quote attributed to Leonard Henry Courtney, in Mark Twain’s autobiography but neither was originating author. It was said by Leonard Henry Courtney a British economist and politician (1832-1918) in a statistics journal no less. The Journal of the Royal Statistical Society, No 59 (1896)

for some reason my comments on this page don’t show up .. i guess the moderator/system doesn’t like me. Glad someone was able to get through with the fact that Mark Twain the quote didn’t originate with him…

My apologize for this. My hosting provider redirected this to a static page when the site was over-run. Unfortunately this somehow wasn’t communicated and puzzled me to no end. I kept approving comments but they did not show up.

Why does the probability of heads-heads increase to 1/3? The combined probability of heads-tails or tails-heads can’t change just because the states can’t be distinguished.

Another way to look at it is to solve for the probability p that the outcome of one “coin”
is “heads”. If the three distinguish states each have equal probability, then p^2 = 2p*(1-p) = (1-p)^2. Solving p^2 = (1-p)^2 for p gives p = 0.5.

It is because nature does not assign identity to the coins when they are bosons. I.e. you can not label a boson. If you exchange them nature counts the resulting state as the very same state. There is no physical meaning to saying that they change position at all if they remain in the same energy quantum state. Hence my reference to nature playing the ultimate shell game.

This was an interesting article, but the last paragraph seems off. I am not sure how Bose-Einstein Condensation (BEC) led to Penning traps. The Penning trap comes from classical E&M and predates the observance of BEC by 35 years. And, true, some of the first laser cooling experiments were performed in Penning traps (1980), but it was that laser cooling that allowed BEC to be realized (1995), not the other way around. I also don’t understand how Penning traps were used in the first realizations of quantum computing (QC). I would have said the first quantum computing realizations occurred in NMR systems and with photons. It is true that trapped ions were used (and are still used) in initial QC experiments, but those were Paul traps, not Penning traps. In any case, it was an interesting article.

Thanks for pointing this out. I cleaned up the last paragraph. The real connection is definitely the laser cooling and the trapping techniques are just a necessary pre-condition for the latter. At any rate for BSC it requires a magnetic trap , so I dropped all the reference to the physicists with P that I always get confused.

I think it’s a bit hard to say when quantum computing started for various systems, since you can really define it any of a number of different ways. I would have thought NMR and photons were doing algorithms before ions were. But if you are talking about gates, then it’s a bit different.

As for BEC and Penning traps, there really isn’t that much of a connection. Penning traps trap charged particles (ions in this case) using a combination of magnetic fields and electric fields that act directly on the charge. And you do not need to laser cool the ions to trap them. Nor do you need quantum mechanics to describe what is going on until you get down to the single phonon level.

A BEC usually involves a magnetic trap that works on neutral atoms rather than ions. It works by having a inhomogeneous field that shifts the atomic energy levels based on physical location, thus pushing the atoms to the point of lowest energy. (This is no at all similar to what the Penning trap does.) Laser cooling is usually used here to help the trapping, but the actual magic of getting to a BEC involves evaporative cooling (where the trap is relaxed, allowing the hottest atoms to escape) not laser cooling. Also, a magnetic trap is only convenient, not necessary, as many BECs are formed in laser dipole traps these days.

I should also point out that the NIST quantum simulator doesn’t involve BEC physics at all. The only connection is that both technologies use laser cooling and involve cold atoms. That means the tools used to perform both experiments are similar, but the underlying physics really isn’t. BEC’s aren’t that useful for quantum computing, since they involve a single wave function, while quantum computing is all about scaling to thousands of quantum systems. But the most closely related QC work would be stuff from people like Trey Porto (also at NIST), Mark Saffman (Wisconsin), Dave Weiss (Penn State), among others.

Again, I liked the article, and learned some interesting things from it!

Fortunately science is not the Olympics, no need to really determine a victor in this race 🙂

The Science article is not really all that clear on this either. Both ion based and NMR were certainly early trail-blazers.

Didn’t mean to touch on a direct BEC quantum computing connection rather make a bit of a tenuous connection back to the main topic of this blog, by pointing out that experimental techniques for ultra cooling have found some application in this field.

On the other hand there are someschemes [PDF] to perform quantum computing with BECs. To my knowledge this is all theoretical at this point though.

Your are certainly entirely correct about the Penning trap, I meant to reference the Paul one but got them mixed up. On second thought though this was taking it a bridge too far, BECs after all, as you mention, requires a trapping mechanism for neutral atoms.

To me the easiest way to think about why nature does not differentiate between the HT and TH state for “boson coins” is to go back to the BE condensate and come to grips what it means that it can be thought of as one coherent quantum state.

We know that it condensated from various bosons but once it forms nature treats it like its on entity. I.e. if you were to pull out two bosons and have them switch places nature doesn’t keep an account of this. Physically it will be exactly the same BEC blob before and after.

In a sense it is as if you reached a kind of “resolution limit” of nature. There is only so many things that nature’s “machinery” can keep track off and the position of identical bosons i.e. occupying the same ground state is not one of them.